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Talk:Bird's number
I kinda skimmed over the source, but where is the number defined in it? FB100Z • talk • 19:27, September 13, 2012 (UTC) * It's defined on page 25. He doesn't mention a name for it, but I've seen other people refer to it as "Bird's number." --Ixfd64 (talk) 02:35, September 14, 2012 (UTC) ::Who's "other people"? FB100Z • talk • 20:23, September 14, 2012 (UTC) :: * See http://www.docstoc.com/docs/70547625/Array-Notations-for-Super-Huge-Numbers---Chris-Bird, there are "Bird's number" in his old notation. It is between {3,3,4 \ 1,2 2} and {3,4,4 \ 1,2 2} in his new notation. By the way it is vastly smaller even than triakulus, which is around {3,3 \ 1 \ 2 2}. Ikosarakt1 (talk) 09:29, September 21, 2012 (UTC) ** Could you please find a mirror so that you don't need to register and pay to download the document? --Cloudy176 (talk) 03:59, November 8, 2012 (UTC) I revised Bird's old notation, his number is larger than I thought, I can state that B ~ {3,3,4 / 2} (Bowers' notation): Firstly, compare versions of old and new Bird's notations: they are identical below hyper-dimensional arrays, but he differently define: {a,b 1,2 2} = {a <{a <{a<...{aa}...>} a> a}> a} (b a's from the centre out), Instead of {a,b 1,2 2} = {a <0,2> b} = {a b} So {a,b 1,2 2} (old notation) corresponds to {a,b,2 1,2 2} (new notation). It makes new notations weaker than old. The next difference is that {a <0 <2> 2> b} = {a b} (b a's, rather than b b's), where the new notation is more powerful (second entry changes as array grows). However, generally, it is comparable. Next observe his --| (old not., later he defines this symbol (negation) as much more powerful) and \ (new not.) symbols, they are work similarly: {3,3 \ 2 2} = {3 <3 <3> 3> 3} {3,3 --| 2 2} = {3 <3 <3> 3> 3} {3,3 \ 1 \ 2 2} = {3 <3 \ <3 <3 \ 3> 3> 3} {3,3 --| 1 --| 2 2} = {3 <3 --| <3 <3 --| 3> 3> 3} Consider the [ 2 ] symbol and \\ (they are similar and make the line of b --|'s and \'s respectively, and might be approximated in BEAF as {a,b,b} & a (\ 2 ~ {a,b,2} & a, \ 1 \ 2 ~ {a,b,3} & a, and so on)), so: {a,b [ [2 ] 2] 2} ~ {a,b \\ 2 2} [ 3 ], [ 4 ], [ 1,2 ] and [1] corresponds to 3\, 4\ and 1,2\ [2\ 2]\ symbols respectively. Secondly, hyper-nested array notation (in new version) corresponds to \( f_{Γ_0} \)(n) in the fast-growing hierarchy, and legions in BEAF at the same. Each additional set of []'s in the old version adds the hyper-nesting level in new not. Knowing this, I guess that Bird's function, X(n) is on par with {n,n,2 / 2}. Next he makes some primitive recursion on this function: X(X(n)) = X^2(n) ~ {n,{n,n,2 / 2},2 / 2} = {n,3,2 / 2} X(X(X(n))) = X^3(n) ~ {n,4,2 / 2} He defines some number N as {3 <<<<<<<3>>>>>>> 3} ~ {3,3,3,3} & 3 & 3 & 3 & 3 & 3 & 3 ~> {3,7 / 2} X(N) >~ {3,{3,7 / 2},2 / 2} > {3,3,3 / 2} X(X(N)) > {3,4,3 / 2} X(X(X(N))) > {3,5,3 / 2} X(X(X(...(X(N)...) X(N) nested layers > {3,{3,3,3 / 2}+2,2 / 2} ~ {3,3,4 / 2} So Bird's number is comparable and larger than the great big boowa. The source is gone I heard that Docstoc is closing down on December this year, so I decided to archive Bird's PDF in it. Unfortunately, the link given in the article already gives a 404. As far as I know the content of the PDF was previously given in a form of a website -- I don't know where it went. We should ask Chris Bird do he still has his old PDF, or the source of the definition of Bird's number might become permanently lost. -- ☁ I want more ⛅ 11:49, November 18, 2015 (UTC) :Nobody cares about Bird's array notation anymore... -- ☁ I want more ⛅ 11:49, November 25, 2015 (UTC) ::Sad but true. LittlePeng9 (talk) 13:49, November 25, 2015 (UTC) :::I searched "Chris Bird" on the search box on the 404 page, and every single document given on the search results gives a 404. Looks like Docstoc is already toast. -- ☁ I want more ⛅ 11:10, November 30, 2015 (UTC) ::::I would also like to ask this question, but I just found an alternative page, in this. Khankao1 (talk) 06:53, March 25, 2016 (UTC) :::::There it is! I knew that Bird formerly had a website describing his notation and allowed the PDF to be downloaded for free, but the site appears to be disappeared. Thankfully there's an archive for it! -- ☁ I want more ⛅ 08:03, March 25, 2016 (UTC) About {a, b 3 2} Does anyone know the bird's old array notation, {a, b 3 2} is equivalent to what? In the above Ikosarakt1 analysis, he will be as it [[3] 2] point of view, but there is no evidence that is the case. I personally think that {a, b 3 2} = {a ‹‹2›› 2} = {a ‹‹1›› 2 2 a ‹‹1›› 2 ... 2 a ‹‹1›› 2}, so that the growth rate is a multilayer brackets can only be achieved \(\varphi(\omega,\omega)\), instead of SVO. Khankao1 (talk) 11:15, April 8, 2016 (UTC)